Of course they say that. People with multiple personalities never declare that they are the same.

lol... point well taken...
The way the poems phrase... there was one in atonment chapter...rika mention that they can't forgive you as they do not know.. but i can forgive you... so maybe frederica is rika but rika is not frederica

Hmm Let us blame Ryukishi07 for having this wonderful twist~ (Its a compliment really)
Anyways let think of other refernces of alternate world
"For example Tsubasa Chronicles different world same people"
If one of those characters would actually met her other version that would be this theory (Frederica = Rika)
In the other way lets take that poem as a whole answer
Rika and Frederica arent the same~

Lol... apparently Frederica and Lambdadelta were fought in a "battle" for around 100 years.
Rika said all this stuff was repeating for around 100 years.
And Frederica looks like Rika and Lambdadelta looks like Miyo.
Maybe they were using Rika and Miyo as pawns.. or something. ♙♟